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I have searched the questions regarding notation of "such that" in this forum but only encountered discussions on the use of : vs. | and less common alternatives.

In some cases I find myself in need of using two "such that" formulations in a nested fashion in set builder notation, consider the following example:

We consider some set of points $\textbf{x} \in \mathcal{X} \subset \mathbb{R}^n$ and some vector of functions $\textbf{r} : \mathbb{R}^n \to \mathbb{R}$.

Now let the subset $\mathcal{F} = \{ \textbf{x} \in \mathcal{X} : \textbf{r}(\textbf{x}) \leq \textbf{0} \}$ be called the set of feasible points (where the inequality is understood to hold pointwise), then its complement could be denoted as $\{ \textbf{x} \in \mathcal{X} : \exists i : r_i(\textbf{x}) > 0 \}$.

While I suppose this notation is valid, it is not very intuitive as it relies on right associativity of the existential quantifier which (at least IMHO) is not very satisfying.

Instead I would actually consider a mixture of notations such as $$\{ \textbf{x} \in \mathcal{X} : \exists i | r_i(\textbf{x}) > 0 \}$$ easier to understand.

  • Is there a downside to this mixed notation that I am unaware of?
  • Is there some common practice for nested such that notation?
mrclng
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    I don't think this is very common, but when it is used you usally write ${ x \in X | (\exists i) r_i(x)>0}$ or also ${ x \in X | (\exists i)[ r_i(x)>0]}$. Also nothing prevents you from writing ${x \in X | \exists i : s.t. : r_i(x)>0 }$. If you use $:$ and $|$ I would intuitively do it the other way around: ${x \in X | \exists i : r_i(x)>0}$. – flawr Oct 08 '17 at 12:17
  • The syntax is ${ x \mid \varphi(x) }$, that reads: "the set of all and only those $x$ such that $\varphi$ holds", where $\varphi(x)$ is a formula with $x$ free, i.e. expressing a "condition" on $x$. – Mauro ALLEGRANZA Oct 08 '17 at 12:32
  • Thsu, we may have: ${ \textbf{x} \in \mathcal{X} : \exists i \ (r_i(\textbf{x}) > 0) }$ which abbreviates: ${ \textbf{x} : \textbf{x} \in \mathcal{X} \land \exists i \ (r_i(\textbf{x}) > 0) }$. The expression $ \textbf{x} \in \mathcal{X} \land \exists i \ (r_i(\textbf{x}) > 0)$ is our $\varphi( \textbf{x} )$. – Mauro ALLEGRANZA Oct 08 '17 at 12:35

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